8
PASCALE CHARPIN
PROPOSITION
2. Let . be a weight of Ey such that . =f. 2m- 1 and suppose
that w(y)
=
.. We denote by "' the dimension of the kernel of the symplectic
form W
y.
Then we have:
{ X
9
y
I
g
E
G }
= {
a
E
Ey
I
w(a)
=
. } ,
and the cardinal of the set above is
2m-~.
So each codeword b of Ey, of weight
different from 2m-
1
is invariant under
2~
translations
LEMMA
1. Let x
E
A\P2
,
y
E
C_i,
Ey
=
y+pm-
1
;
Dx is the code CxUC,
where Cx is the coset x +C. Then
v;-
contains half of elements of Ey:
card E
card { a E Ey
I
x, a
=
0 }
=
2 Y
=
2m .
PROPOSITION
3. Let x
E
A\P2

Let. be a weight of Ey and suppose that
w(y)
=
.. We denote by
N.
the number of codewords of Ey of weight.. Set
i~. =card { a
E
Ey
I
w(a)
=
. and x, a=
0 } .
Then
X E
'P\'P2
==?
if).,
=
N2m-)..
X
E
A\P
==?
N)..
=
N).. -
N2m-).. .
Moreover, if .=f. 2m-
1
we have:
(11)
By hypothesis the code Cl_ is an ideal of A, since it is invariant under any
translation. Thus the cosets of C of minimum weight 1 have the same weight
distribution. By applying Formula {11), we can compute this weight distribution.
CoROLLARY
1. Suppose that the dual of the code C consists of the code pm- 1
itself and L sets of cosets of same type; that is: Li cosets of type (hi),
i
E [1, L].
Let
x
=
XY, g
E
G and Dx
=
(x
+C)
U
C. Set
Al
=card { c E
v;
I
w(c)
=
l} and .i
=
2m-
1
-
2m-h;-
1
'
i
E [1,L].
So the weights of
v;-
are elements of the set { 0, 2m-1, .i, 2m-.i (i E [1,
r]) }.
Then the coefficients of the weight polynomial of
v;-
are, for each
i
in
[1,
L]:
Ao
=
1 and A2m_
1
=
/3/2 where
f3
is the number of codewords of weight 2m-
1
in Cl_. The weight distribution of the coset x
+
C can be calculated from the
Formula (10}.
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